A general theorem in spectral extremal graph theory
John Byrne, Dheer Noal Desai, Michael Tait
TL;DR
This work develops a general theorem connecting edge-extremal and spectral-extremal graphs in spectral extremal graph theory, under the condition ex(n, F)=O(n). It shows that spectral extremality tends to enforce a K_{k,n−k}–like structure and, in many cases, forces SPEX(n, F) to coincide with EX(n, F) or lie within its edge-extremal family; it further extends these insights to the alpha-spectral setting (SPEX_α). The authors provide a detailed eigenweight analysis and structural decomposition, yielding broad applications to families such as paths, matchings, linear forests, trees, even and disjoint cycles, minors, and topological subdivisions, recovering numerous known results and proving new ones. The paper closes with a discussion of limitations via a counterexample, and outlines open questions and potential relaxations for further generalization.
Abstract
The extremal graphs $\mathrm{EX}(n,\mathcal F)$ and spectral extremal graphs $\mathrm{SPEX}(n,\mathcal F)$ are the sets of graphs on $n$ vertices with maximum number of edges and maximum spectral radius, respectively, with no subgraph in $\mathcal F$. We prove a general theorem which allows us to characterize the spectral extremal graphs for a wide range of forbidden families $\mathcal F$ and implies several new and existing results. In particular, whenever $\mathrm{EX}(n,\mathcal F)$ contains the complete bipartite graph $K_{k,n-k}$ (or certain similar graphs) then $\mathrm{SPEX}(n,\mathcal F)$ contains the same graph when $n$ is sufficiently large. We prove a similar theorem which relates $\mathrm{SPEX}(n,\mathcal F)$ and $\mathrm{SPEX}_α(n,\mathcal F)$, the set of $\mathcal F$-free graphs which maximize the spectral radius of the matrix $A_α=αD+(1-α)A$, where $A$ is the adjacency matrix and $D$ is the diagonal degree matrix.